A representative relationship obtains just one object from the linked in objects.
This is best explained with an example. Suppose you have a term view that shows you the terms in your vocabulary: Horse, Cat, Aardvark.
In addition to the term names, you want to show the title of the most recent node in that term. This would give you a view that shows: Horse - latest horse node title, Cat - latest cat node title, Aardvark - latest aardvark node title.
Each of these is the title of the representative node for that term, chosen by creation time.
Any sort criterion can be used to choose the representative node. You might instead want to show the node with the most comments for each term, the first node in alphabetical order, or if you have a voting module installed, the most popular node.
The options for a representative relationship let you choose a sort criterion and a sort order. This determines how the representative object is chosen: the first object returned by the sort is shown. For example, choose 'Node: title' and 'Ascending' to get the first node by title as the representative; or 'Node: comment count' and 'Descending' to get the node with the most comments.

Performance

These relationships require a correlated subquery. This can be slow to run on large amounts of data, as the subquery must be run for every row of the main query.
For more on this topic, see the MySQL tutorial on group-wise maximum queries.